Abstract
The chapter proposes structural condition monitoring for buildings and mechanical structures using a new nonlinear filtering method under the name Derivative-Free Nonlinear Kalman Filtering. The filter makes use of exact linearization of the structure's dynamical model in accordance to differential flatness theory and of an inverse transformation that enables one to obtain estimates for the state vector elements of the initial model. The response of the structure is compared to the response generated by the filter under the assumption of a damage-free model. Moreover, the filter provides estimates of the state vector elements of the structure, which cannot be directly measured, while it can also give estimates of unknown excitation inputs. By comparing the two signals, residuals sequences are generated. The statistical processing of the residuals provides an indication about the existence of parametric changes (damages) in the structure that otherwise could not have been detected.
Top1. Introduction
Damages in civil or mechanical structures cannot always be detected by visual inspection. Therefore, it is necessary to develop elaborated methods for structural health monitoring that will be capable of diagnosing the existence of parametric changes and failures (Wu, Zhou, & Yang, 2006), (Campillo & Mevel., 2005). Previous results in the area of Kalman Filter-based structural condition monitoring can be found in (Lei, Wu.& Li, 2012), (Lei, Jiang & Xu, 2012), (Zhou, 2008), (Katkhuda & Hadlar, 2008). In the present chapter, the development of a new structural health monitoring approach is proposed making use of nonlinear Kalman Filtering and of statistical signal processing.
The multi-DOF building or mechanical structure is modeled as a set of coupled nonlinear oscillators that can be subjected to external excitation. First, it is shown that the dynamic model of the monitored structure is differentially flat and admits static feedback linearization (Sira-Ramirez and Agrawal, 2004), (Rigatos, 2011), (Fliess & Mounier, 1999), (Rouchon, 2005), (Bououden, Boutat, Zheng, Barbot, & Kratz, 2011). It is also shown that the dynamic model can be written in the linear Brunovsky canonical form for which a state feedback controller can be easily designed, thus also enabling active control of the structure. Next, to estimate the building’s or mechanical structure’s motion characteristics from indirect measurements a new nonlinear filtering method under the name Derivative-free nonlinear Kalman Filtering is introduced (Rigatos, 2012a), (Rigatos, 2012b), (Rigatos & Tzafestas, 2007). The considered filter makes use of the aforementioned exact linearization transformation of the structure’s dynamical model, in accordance to differential flatness theory, and of an inverse transformation that enables to obtain estimates for the state vector elements of the initial model. The filter provides estimates of the state vector elements of the structure which cannot be directly measured. Moreover, by redesigning the Derivative-free nonlinear Kalman Filter as a disturbance observer, it is shown that it is possible to estimate unknown external inputs applied to the structure.
The response of the structure is recorded through suitable sensors (in the form of a wireless sensors network deployed at specific measurement points) and is compared against the response generated by nonlinear Kalman Filtering under the assumption of a damage-free model (to improve accuracy it is possible to apply a sensor fusion-based implementation of the Kalman Filter). By comparing the two signals, residuals sequences are generated. The processing of the residuals with the use of statistical decision making criteria provides an indication about the existence of parametric changes (damages) in the structure, which otherwise could not have been detected (Basseville & Nikiforov, 1993), (Zhang, Basseville & Benveniste, 1998), (Rigatos. & Zhang, 2009). Thus, by applying fault detection tests based on the 
distribution it can be concluded if the structure remains healthy and if the nominal parameter values for its model still hold. Otherwise, a failure can be detected. Moreover by applying the 
tests in compartments of the building’s model, the faulty substructure can be also isolated.
Key Terms in this Chapter
Fault Isolation: An information processing method that enables to identify which is the component or parameter of the system that is responsible for the symptoms of the faulty behavior.
H0: The system’s parameters maintain their nominal values.
Nonlinear Estimation: A computational method which calculates non-measurable features (e.g. parameters or state vector elements) of a nonlinear time-varying or time-invariant model, through the processing of information coming from a small number of measuring units (e.g. sensors). The method is based on the sequential adaptation of the estimates of the non-measurable variables by suitably computed coefficients which are based on the estimation error (e.g. difference between the real and the measured system’s output).
H1: The system’s parameters have changed from their nominal values. The critical variable for the test is the weighted square of another variable that follows the Gaussian distribution (estimation error of the state vector of the monitored structure).
Brunovsky Canonical Form: A state space description of the system in which the state vector elements are connected to each other through a chained integration procedure (state variable is the integral of state variable ) while the last state variable is equal to the integral of the control input.
Coupled Nonlinear Oscillators: Masses linked through elastic parts (e.g. springs) and absorbers (e.g. dampers) which when excited by an external mechanical force exhibit an oscillatory motion. The differential equation describing the motion of each mass is a nonlinear one, while the displacement of each mass affects, through coupling terms, the motion of neighboring masses.
Wireless Sensors Network: A set of spatially distributed sensors, which exchange information either between them or with a central information processing unit. In the case of structural condition monitoring such a sensors network provides information about the motion or deformation characteristics of each floor of the building.
Differential Flatness Theory: A mainstream area in nonlinear control systems theory, which analyzes dynamical systems after considering that all state variables and control inputs of the associated dynamical model can be expressed as functions of one single algebraic variable, the so-called “flat output” and also as functions of the flat-output’s derivatives. Differential flatness theory can cope efficiently with the control and state estimation of complicated nonlinear dynamical systems of the lumped parameter and distributed parameter type.
Fault Detection: An information processing method that enables to decide whether the monitored system has undergone a fault (deviation of its structural elements from their nominal values) or not.
Exact Linearization Transformation: A procedure for transforming a nonlinear dynamical system into an equivalent linear state-space description that is based on a change of variables (diffeomorphism) and which does not introduce any numerical errors. It is distinguished from approximative linearization round local equilibria which is based on a Taylor series expansion and the computation of Jacobian matrices. In the latter approach, the truncation of higher order terms from the series expansion introduces numerical errors.
Multi-DOF Building or Mechanical Structure: A building or mechanical structure exhibiting motion of its parts in various directions (degree’s of freedom). For example in an n-storey building motion, each floor exhibits its own motion characteristics and is associated with one degree of freedom (DOF).
Disturbance Observer: An observer that estimates the extended state vector of a dynamical system, the latter comprising as additional elements the external disturbances and their derivatives.
X2 Change Detection Test: A hypothesis test based on the X 2 distribution. It consists of two hypotheses.
Residuals: A sequence of variables representing the evolution in time of the difference between the monitored structure characteristics and the associated outputs estimated with the use of the Kalman Filter.
Derivative-Free Nonlinear Kalman Filter: A nonlinear filtering algorithm which consists of the Kalman Filter recursion on the equivalent model of the system, which is obtained after applying a transformation based on differential flatness theory. It also makes use of an inverse transformation, based again on differential flatness theory, that enables to obtain estimates for the state variables of the initial nonlinear model.
Structural Condition Monitoring: Information processing methods aiming at detecting the existence of failures in structures (mainly buildings and mechanical constructions).
Static Feedback Linearization: Procedure for the linearization of the system dynamics through a change of coordinates (transformation of its state variables) and feedback of exclusively the system’s state vector. It is distinguished from dynamic feedback linearization in which the state vector is extended by considering as additional elements the control inputs and their derivatives.